topology of de nable groups in tame structuresberardu/art/paris2011/paris.pdf · topology of de...

Topology of definable groups

in tame structures

Alessandro Berarduccijoint work with Elias Baro

Geometrie et Theorie des Modeles

Paris, Ecole Normale Superieure

6 Mai 2011

() May 4, 2011 1 / 32

O-minimal structures

Definition

A structure M = (M, <, . . .) is o-minimal if every definable subset of M isa finite union of points and intervals (a, b) with a, b ∈ M ∪ {±∞}.

Example

1 (R, <,+, ·).

2 Any real closed field (R, <,+, ·).

3 (R, <,+, ·, exp, sin|[0,1])

() May 4, 2011 2 / 32

Triangulations

Fix an o-minimal structure M. We always assume that M expands anordered field. We recall the following basic result.

Theorem ([vdD98])

Every definable set X ⊂ Mn can be triangulated, namely there is a finite(open) simplicial complex K and a definable homeomorphismf : |K |M → X .

In general two compact polyhedra |K | and |L| can be homeomorphicwithout being PL-homeomorphic. However:

Theorem (O-minimal Hauptvermutung: [Shi10])

Let K , L be finite closed simplicial complexes and let f : |K |M → |L|M be adefinable homeomorphism. Then K and L have isomorphic subdivisions.So there is a PL-homeomorphism g : |K |M ∼= |L|M .

() May 4, 2011 3 / 32

Triangulations

Theorem ([vdD98])

In general two compact polyhedra |K | and |L| can be homeomorphicwithout being PL-homeomorphic.

However:

() May 4, 2011 3 / 32

Triangulations

Theorem ([vdD98])

In general two compact polyhedra |K | and |L| can be homeomorphicwithout being PL-homeomorphic. However:

() May 4, 2011 3 / 32

Definable groups

Definition

A definable group is a definable set G ⊂ Mn with a definable groupoperation.

Example

SO2(M) =

{(a −bb a

): a2 + b2 = 1

Example

A non singular elliptic curve y 2 = x3 + ax + b in P2(M) (“real” case) or inP2(M[

√−1]) (“complex” case).

() May 4, 2011 4 / 32

Definable groups

Definition

Example

SO2(M) =

{(a −bb a

): a2 + b2 = 1

Example

() May 4, 2011 4 / 32

Definable groups

Definition

Example

SO2(M) =

{(a −bb a

): a2 + b2 = 1

Example

() May 4, 2011 4 / 32

Lie tori

Every connected compact abelian real Lie group is Lie-isomorphic to atorus Tn = Rn/Zn.

In the o-minimal context Z does not exist, but we can define:

Definable tori

Let T1(M) = [0, 1) ⊂ M with the following group operation:

x ∗ y =

{x + y if x + y < 1x + y − 1 otherwise

Similarly we define the n-th torus Tn = [0, 1)n.

() May 4, 2011 5 / 32

Lie tori

In the o-minimal context Z does not exist,

but we can define:

Definable tori

x ∗ y =

() May 4, 2011 5 / 32

Lie tori

In the o-minimal context Z does not exist, but we can define:

Definable tori

x ∗ y =

() May 4, 2011 5 / 32

t-topology

Theorem ([Pil88])

Any definable group (G , ∗) admits a unique group topology, called thet-topology, such that G is a finite union U1 ∪ . . . ∪ Un with each Ui openin G and definably homeomorphic to Mn (n = dim(G )).

So when M = (R, <, . . .), any definable group G is a real Lie group.

Example

With the t-topology the torus T1 = [0, 1) (with addition modulo 1) isdefinably homeomorphic to S1 (the unit circle in M), so the t-topologydoes not coincide with the subspace topology [0, 1) ⊂ R.

() May 4, 2011 6 / 32

t-topology

Theorem ([Pil88])

Any definable group (G , ∗) admits a unique group topology, called thet-topology, such that G is a finite union U1 ∪ . . . ∪ Un with each Ui openin G and definably homeomorphic to Mn (n = dim(G )).

So when M = (R, <, . . .), any definable group G is a real Lie group.

Example

With the t-topology the torus T1 = [0, 1) (with addition modulo 1) isdefinably homeomorphic to S1 (the unit circle in M), so the t-topologydoes not coincide with the subspace topology [0, 1) ⊂ R.

() May 4, 2011 6 / 32

t-topology

Definition

G is definably compact if every definable curve f : (0, ε)→ G has a limitin G in the t-topology.G is definably connected if G has no proper definable clopen subset inthe t-topology. This is equivalent to say that G has not definablesubgroups of finite index.

By “Robson’s embedding theorem” we have:

Every definable group G can be embedded in some Mm, namely G isdefinably isomorphic to a group G ′ ⊂ Mm such that the t-topology of G ′

coincides with the subspace topology inherited from Mm.

Such a G ′ is definably compact iff it is closed and bounded in Mm.

() May 4, 2011 7 / 32

t-topology

Definition

G is definably compact if every definable curve f : (0, ε)→ G has a limitin G in the t-topology.G is definably connected if G has no proper definable clopen subset inthe t-topology. This is equivalent to say that G has not definablesubgroups of finite index.

By “Robson’s embedding theorem” we have:

Every definable group G can be embedded in some Mm, namely G isdefinably isomorphic to a group G ′ ⊂ Mm such that the t-topology of G ′

coincides with the subspace topology inherited from Mm.

Such a G ′ is definably compact iff it is closed and bounded in Mm.

() May 4, 2011 7 / 32

One-dimensional definable groups

We have seen various examples of one-dimensional definable groups:SO(2,M), T1(M), elliptic curves in P2(M).

Note that SO(2,R) is Lie-isomorphic to T1(R) but not definablyisomorphic in (R, <,+, ·) (i.e. the isomorphism is not semialgebraic).

Theorem ([Str94b])

Let G be a definably compact definably connected abelian definable groupin M. Assume dim(G ) = 1. Then G , with the t-topology, is definablyhomeomorphic to T1 (equivalently: to the unit circle S1).

A natural conjecture is that when dim(G ) = n, then G is definablyhomeomorphic to Tn. The difficulty is that G may have noone-dimensional definable subgroups ([PS99]).

() May 4, 2011 8 / 32

Theorem ([Str94b])

() May 4, 2011 8 / 32

Theorem ([Str94b])

() May 4, 2011 8 / 32

Theorem ([Str94b])

() May 4, 2011 8 / 32

Results

Theorem ([BB11])

Let G be a definably compact definably connected abelian n-dimensionaldefinable group with n 6= 4. Then G is definably homeomorphic to Tn. Inthe semialgebraic case the proviso n 6= 4 is not needed.

Main steps.

1 The result holds in the semialgebraic case;

2 Homotopy transfer;

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

4 πk(G ) = 0 for k > 0;G is definably homotopy equivalent to Tn ([BMO10]);

5 If dim(G ) 6= 4, there is a finite cover f : G → Tn (as spaces). Use fto put a semialgebraic group operation on dom(G ).

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Results

Theorem ([BB11])

Main steps.

3 π1(G ) ∼= Zn ∼= π1(Tn) ([EO04]);

() May 4, 2011 9 / 32

Step 1: Semialgebraic case

Lemma (Elimination of parameters)

Let G be a semialgebraic group over a real closed field M. Then G issemialgebraic homeomorphic (but not necessarily isomorphic) to a

semialgebraic group over Qreal ≺ M.

Proof.

We can assume that the t-topology of G is the topology inheritedfrom the ambient space Mm (Robson’s embedding theorem).

By the triangulation theorem we can assume that dom(G ) = |K |where K is a finite simplicial complex definable without parameters.

However the group operation may need parameters from M.

Since Qreal ≺ M, there is a possibly different commutative group

operation ⊕ on |K | which is defined over Qreal.

() May 4, 2011 10 / 32

Lemma (Elimination of parameters)

Let G be a semialgebraic group over a real closed field M. Then G issemialgebraic homeomorphic (but not necessarily isomorphic) to a

semialgebraic group over Qreal ≺ M.

Proof.

We can assume that the t-topology of G is the topology inheritedfrom the ambient space Mm (Robson’s embedding theorem).

By the triangulation theorem we can assume that dom(G ) = |K |where K is a finite simplicial complex definable without parameters.

However the group operation may need parameters from M.

Since Qreal ≺ M, there is a possibly different commutative group

operation ⊕ on |K | which is defined over Qreal.

() May 4, 2011 10 / 32

Theorem ([BB11])

Let G be a semialgebraic group of dimension n over a real closed field M.Suppose G is definably compact, definably connected, abelian. Then G isdefinably homeomorphic to Tn(M).

Proof.

We can assume that G is defined over Qreal ≺ M. Consider G (R).

G (R) is a compact connected abelian Lie group with the t-topology.

There is an analytic isomorphism h : G (R)→ Tn(R).h is definable in the o-minimal structure Ran.

By Shiota’s o-minimal Hauptvermutung there is a semialgebraichomeomorphism f : G (R)→ Tn(R).

Since Qreal ≺ M we can take f defined without parameters.

The same formula gives a definable homeomorphismf M : G (M)→ Tn(M).

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

There is an analytic isomorphism h : G (R)→ Tn(R).

h is definable in the o-minimal structure Ran.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Theorem ([BB11])

Proof.

() May 4, 2011 11 / 32

Step 2: Homotopy transfer

Recall that homotopy equivalence is much weaker than homeomorphism.

Example

The figure “8” is homotopy equivalent to R2 minus two points.

The homotopy category is more flexible than the topological category.

|K | and |L| are homotopy equivalent if and only if they aresemialgebraically homotopy equivalent.

But two polyhedra |K | and |L| can be homeomorphic without beingdefinably homeomorphic in any o-minimal expansion of R.

() May 4, 2011 12 / 32

Recall that homotopy equivalence is much weaker than homeomorphism.

Example

The figure “8” is homotopy equivalent to R2 minus two points.

The homotopy category is more flexible than the topological category.

|K | and |L| are homotopy equivalent if and only if they aresemialgebraically homotopy equivalent.

But two polyhedra |K | and |L| can be homeomorphic without beingdefinably homeomorphic in any o-minimal expansion of R.

() May 4, 2011 12 / 32

Let M be an o-minimal expansion of a field, and let X be a semialgebraic

set defined over Qreal ⊂ M. Look at X (R) and X (M).

Theorem (Homotopy transfer)

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

3 H∗def (X (M)) ∼= H∗(X (R)) (duality plus [EO04]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

The superscript “def” refers to the relativization to the definable categoryand will be omitted in the sequel.

Failure of topological transfer (see [BO03, Shi10, BB11])

If |K |(M) is a definable manifold, then |K |(R) is a (PL) manifold. But if|K |(R) is a topological manifold, |K |(M) need not be a definable manifold.

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

1 π1def (X (M)) ∼= π1(X (R)) ([BO02]);

2 H∗def (X (M)) ∼= H∗(X (R)) ([BO02]);

4 πndef (X (M)) ∼= πn(X (R)) ([BO09]).

() May 4, 2011 13 / 32

Proposition

If X is a definable set, π1(X ) is finitely generated. Similarly Hn(X ) andHn(X ) are finitely generated for all n.

Proof.

By homotopy transfer after triangulating X

However πn(X ) may not be finitely generated for n > 1.

Example

Let X = S1 ∧ S2. Then π2(X ) = Z(ω).

Indeed, for n > 1, πn(X ) = πn(X ) where X is the universal cover of X .When X = S1 ∧ S2 the space X is an infinite line with infinitely manycopies S2 attached to it, so clearly π2(X ) is not finitely generated as agroup (however it is finitely generated as a Z[π1(X )]-module).

() May 4, 2011 14 / 32

Proposition

Proof.

Example

Let X = S1 ∧ S2. Then π2(X ) = Z(ω).

() May 4, 2011 14 / 32

Proposition

Proof.

Example

Let X = S1 ∧ S2. Then π2(X ) = Z(ω).

() May 4, 2011 14 / 32

Step 3: Torsion

Theorem ([Str94a])

Let G be a definable abelian group. Then the k-torsion subgroup G [k] isfinite.

Question ([PS99])

Suppose G is definably compact. Is G [k] non-empty?

Theorem ([EO04])

Let G be a definably compact definably connected abelian group ofdimension n. Then G [k] ∼= Tn[k].

The proof depends on the study of π1(G ). One shows thatπ1(G ) ∼= π1(Tn) and that G [k] ∼= π1(G )/kπ1(G ). So we need to studyπ1(G ).

() May 4, 2011 15 / 32

Step 3: Torsion

Theorem ([Str94a])

Question ([PS99])

Theorem ([EO04])

() May 4, 2011 15 / 32

Step 3: Torsion

Theorem ([Str94a])

Question ([PS99])

Theorem ([EO04])

() May 4, 2011 15 / 32

Step 3: Fundamental group

Theorem ([EO04])

Let G be a definably compact definably connected abelian group ofdimension n. Then π1(G ) ∼= Zn.

Proof.

Since dom(G ) is a definable set, π1(G ) is finitely generated.

Since G is abelian, the map pk : G → G , x 7→ kx , is ahomomorphisms.

It is also a definable covering map, so it induces an injectivehomomorphism pk∗ : π1(G )→ π1(G ), given by [γ] 7→ k[γ].

Since this holds for every k , π1(G ) is torsion free.

Being also abelian and finitely generated, π1(G ) ∼= Zs for some s.

The proof of s = n uses the cup product in cohomology plus thetheory of Hopf-spaces (details omitted).

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Theorem ([EO04])

Proof.

() May 4, 2011 16 / 32

Step 4: Higher homotopy groupsWe have seen that if X is a definable set πm(X ) may not be finitelygenerated. However:

Lemma ([BMO10])

If G is a definably connected definable group in M, then πm(G ) is finitelygenerated for all m.

Proof.1 If G is a real Lie group, then the fundamental group π1(G) acts

trivially on πm(G) for m > 1, namely G is a simple space.

2 (Serre 1953) If X is a simple space and Hm(X ) is finitely generatedfor all m, then πm(X ) is finitely generated for all m.

3 Being a definable group, our group G is a simple space in thedefinable category.

4 Conclude by a suitable homotopy transfer.

() May 4, 2011 17 / 32

Lemma ([BMO10])

Proof.

1 If G is a real Lie group, then the fundamental group π1(G) actstrivially on πm(G) for m > 1, namely G is a simple space.

() May 4, 2011 17 / 32

Lemma ([BMO10])

() May 4, 2011 17 / 32

Lemma ([BMO10])

() May 4, 2011 17 / 32

Lemma ([BMO10])

() May 4, 2011 17 / 32

Lemma ([BMO10])

() May 4, 2011 17 / 32

Step 4: Higher homotopy groups

In the abelian case we get:

Corollary

Let G be a definably connected definable abelian group in M. Thenπm(G ) = 0 for all m > 1.

Proof.

The morphism pk : G → G , x 7→ kx , is a covering map, so it inducesa injective endomorphism of πm(G ) given by multiplication by k .

Since this holds for all k, we deduce that πm(G ) is divisible.

Since it is also abelian and finitely generated, it must be zero.

In particular this gives a proof that πm(Tn) = 0 for m > 0 withoutfactoring Tn into one-dimensional subgroups.

() May 4, 2011 18 / 32

Corollary

Proof.

() May 4, 2011 18 / 32

Corollary

Proof.

() May 4, 2011 18 / 32

Theorem ([BMO10])

Let G be a definably connected definably compact definable abelian groupin M. Then G is definably homotopy equivalent to Tn(M).

Proof.

By [EO04], π1(G ) ∼= Zn.

Consider the map f : Tn → G sending (t1, . . . , tn) ∈ [0, 1)n toγ1(t1) + . . .+ γn(tn) where [γ1], . . . , [γn] are free generators of π1(G ).

Then clearly f∗ : π1(Tn) ∼= π1(G ).

Since πm(G ) = 0 for m > 1, f induces an isomorphism on all theπm’s.

By the o-miniamal version of Whitehead’s theorem ([BO09]) f is adefinable homotopy equivalence.

() May 4, 2011 19 / 32

Theorem ([BMO10])

Let G be a definably connected definably compact definable abelian groupin M. Then G is definably homotopy equivalent to Tn(M).

Proof.

By [EO04], π1(G ) ∼= Zn.

Consider the map f : Tn → G sending (t1, . . . , tn) ∈ [0, 1)n toγ1(t1) + . . .+ γn(tn) where [γ1], . . . , [γn] are free generators of π1(G ).

Then clearly f∗ : π1(Tn) ∼= π1(G ).

Since πm(G ) = 0 for m > 1, f induces an isomorphism on all theπm’s.

By the o-miniamal version of Whitehead’s theorem ([BO09]) f is adefinable homotopy equivalence.

() May 4, 2011 19 / 32

Step 5: Homotopy tori

We have seen that our group G is definably homotopy equivalent toTn(M) and we want to prove that it is definably homeomorphic to it.

We could try to transfer “Borel’s conjecture” from R to M:

Borel’s conjecture

Let X be a PL-manifold homotopy equivalent to Tn(R) (considered as aPL-manifold under a standard triangulation). Then X is PL-homemorphicto Tn(R).

... but unfortunately this conjecture is false (it turns out that X is alwayshomeomorphic to Tn(R) but the homomorphism is not necessarily PL).

() May 4, 2011 20 / 32

As a partial substitute for Borel’s conjecture we have:

Theorem (Homotopy tori)

Let X be a (closed) PL-manifold of dimension n 6= 4 homotopy equivalentto Tn(R). Then there is a finite PL-covering f : Tn(R)→ X .

References

For n ≥ 5, see [HW69]. The case n = 3 follows from [KS77] plus thepositive solution of Poincare’s conjecture. For n ≤ 2 X is alreadyPL-homeomorphic to Tn(R).

() May 4, 2011 21 / 32

As a partial substitute for Borel’s conjecture we have:

Theorem (Homotopy tori)

Let X be a (closed) PL-manifold of dimension n 6= 4 homotopy equivalentto Tn(R). Then there is a finite PL-covering f : Tn(R)→ X .

References

For n ≥ 5, see [HW69]. The case n = 3 follows from [KS77] plus thepositive solution of Poincare’s conjecture. For n ≤ 2 X is alreadyPL-homeomorphic to Tn(R).

() May 4, 2011 21 / 32

By triangulating our group G and transfering from R to M the result onHomotopy tori we get:

There is a semialgebraic (even PL) finite cover

f : Tn(M)→ G (M)

(as spaces, not as groups).

This is nice, but it would have been more useful to have a cover going theother way around.

() May 4, 2011 22 / 32

Let G be definably compact, definably connected, abelian, of dim 6= 4.Then there is a definable finite cover h : G (M)→ Tn(M).

Proof.

Start with the finite cover f : Tn(M)→ G (M) (as spaces).

Lift the group operation on G to a definable group operation ∗ on Tn.So f becomes a cover of definable groups f : H = (Tn, ∗)→ G .

Since ker(f ) < H is finite, ker(f ) < H[m] for some m.

So we get a group cover G ∼= H/ker(f )→ H/H[m].

H is a definable abelian group, so by [Str94a] H[m] is finite. Deducethat H/H[m] ∼= H.

So we get a definable cover h : G (M)→ Tn(M).

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Proof.

Lift the group operation on G to a definable group operation ∗ on Tn.

So f becomes a cover of definable groups f : H = (Tn, ∗)→ G .

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Proof.

() May 4, 2011 23 / 32

Step 5: Reduction to the semialgebraic case

Let G be definably compact, definably connected, abelian, of dim 6= 4.Then G is definably homeomorphic (not isomorphic!), to a semialgebraicabelian group.

Proof.

We can assume that dom(G ) = |K | (triangulation).

We have a definable cover h : G → Tn (as spaces).

Since Tn is semialgebraic, there is a semialgebraic cover h′ : G ′ → Tn

and a definable homeomorphism φ : G ≈ G ′ with h′ ◦ φ = h.(see [EJP10])

By the o-minimal Hauptvermutung there is a semialgebraichomeomorphism ψ : G ≈ G ′ and therefore h′ ◦ ψ : G → Tn is asemialgebraic cover.

Tn admits a semialgebraic commutative group operation which can belifted to G via the cover.

() May 4, 2011 24 / 32

Proof.

() May 4, 2011 24 / 32

Proof.

() May 4, 2011 24 / 32

Proof.

() May 4, 2011 24 / 32

Proof.

() May 4, 2011 24 / 32

Conclusions

Let G be a definably connected definably compact definable abelian groupin M. Even in dim = 4 we have:

Corollary

Suppose dom(G ) = |K |(M). Then |K |(R) is homeomorphic to Tn(R).

Proof.

By [FQ90] a PL-manifold homotopy equivalent to Tn(R) is homeomorphicto Tn(R).

However in principle the homeomorphism may be wild (not PL), so wecannot deduce |K |(M) is definably homeomorphic to Tn(M).

() May 4, 2011 25 / 32

Conclusions

A possible approach to deal with the case dim(G ) = 4 is to replace G withG × T1 (a group of dim = 5). But ...

Question

Let X be a semialgebraic set and suppose that X (R)× T1(R) issemialgebraically homeomorphic to T5(R). Is X (R) semialgebraicallyhomeomorphic to T4(R)?

The answer is yes if T1 is replaced by R (Shiota). With T1 there could beproblems.

() May 4, 2011 26 / 32

Conclusions

Question

() May 4, 2011 26 / 32

Conclusions

Question

() May 4, 2011 26 / 32

Conclusions

Another possibility to deal with the dim = 4 case is to use the work ofmany authors on the “infinitesimal subgroup” G 00 of G (see[Pil04, BOPP05]):

Theorem1 If G is a definable group in M and M∗ � M is saturated, then

G = G (M∗)/G 00(M∗) is a compact real Lie group [BOPP05].

2 When G is definably compact, G 00 is torsion free anddimM(G ) = dimR(G)[HPP07].

When G is definably compact, definably connected and abelian, it followsthat G ∼= Tn(R). We could try to transfer results from G to G and deducethat G is definably homeomorphic to Tn(M).

() May 4, 2011 27 / 32

Conclusions

Theorem (Transfer from G to G )

Let G be definably compact and let p : G → G = G (M∗)/G 00(M∗) be theprojection.

1 dimM(G ) = dimR(G) [HPP07].

2 The image under p of a nowhere dense definable X ⊂ G , has measurezero in G. [HP09].

3 If U ⊂ G is open and V ⊂ G is the preimage of U, thenπ1(U) ∼= πdef1 (V ) [BM11].

4 G determines the definable homotopy type of G [Bar09, BM11].

Taking G abelian and U = G in (3) we obtain π1(G ) ∼= π1(G) (∼= Zn).However (3) uses (2), which in turn uses π1(G ) ∼= Zn.

Conjecture

If G is triangulated by |K |(M), then G can be triangulated by |K |(R).

() May 4, 2011 28 / 32

Conclusions

Theorem (Transfer from G to G )

Let G be definably compact and let p : G → G = G (M∗)/G 00(M∗) be theprojection.

1 dimM(G ) = dimR(G) [HPP07].

2 The image under p of a nowhere dense definable X ⊂ G , has measurezero in G. [HP09].

3 If U ⊂ G is open and V ⊂ G is the preimage of U, thenπ1(U) ∼= πdef1 (V ) [BM11].

4 G determines the definable homotopy type of G [Bar09, BM11].

Taking G abelian and U = G in (3) we obtain π1(G ) ∼= π1(G) (∼= Zn).However (3) uses (2), which in turn uses π1(G ) ∼= Zn.

Conjecture

If G is triangulated by |K |(M), then G can be triangulated by |K |(R).

() May 4, 2011 28 / 32

References I

[Bar09] Elıas Baro.On the o-minimal LS-category.To appear in: Israel Journal of Mathematics, 2009:1–14, 2009.

[BB11] Elıas Baro and Alessandro Berarducci.Topology of definable abelian groups in o-minimal structures.arXiv: 1102.2494v1, pages 1–7, 2011.

[BM11] Alessandro Berarducci and Marcello Mamino.On the homotopy type of definable groups in an o-minimal structure.Journal of the London Mathematical Society, DOI:10.111:1–24, February 2011.

[BMO10] Alessandro Berarducci, Marcello Mamino, and Margarita Otero.Higher homotopy of groups definable in o-minimal structures.Israel Journal of Mathematics, 180(1):143–161, October 2010.

[BO02] Alessandro Berarducci and Margarita Otero.O-minimal fundamental group, homology and manifolds.J. London Math. Soc., 2(65):257–270, 2002.

[BO03] Alessandro Berarducci and Margarita Otero.Transfer methods for o-minimal topology.Journal of Symbolic Logic, 68(3):785–794, 2003.

() May 4, 2011 29 / 32

References II

[BO09] Elıas Baro and Margarita Otero.On O-Minimal Homotopy Groups.The Quarterly Journal of Mathematics, 61(3):275–289, March 2009.

[BOPP05] Alessandro Berarducci, Margarita Otero, Ya’acov Peterzil, and Anand Pillay.A descending chain condition for groups definable in -minimal structures.Annals of Pure and Applied Logic, 134(2-3):303–313, July 2005.

[EJP10] Mario J. Edmundo, Gareth O. Jones, and Nicholas J. Peatfield.Invariance results for definable extensions of groups.Archive for Mathematical Logic, 50(1-2):19–31, July 2010.

[EO04] Mario J. Edmundo and Margarita Otero.Definably compact abelian groups .Journal of Mathematical Logic, 4(2):163–180, 2004.

[FQ90] M.H. Freedman and F. Quinn.Topology of 4-manifolds.Princeton Mathematical Series(39), Princeton University Press, Princeton, NJ, 1990.

[HP09] Ehud Hrushovski and Anand Pillay.On NIP and invariant measures.Preprint, pages 1–69, 2009.

() May 4, 2011 30 / 32

References III

[HPP07] Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay.Groups, measures, and the nip.Journal of the American Mathematical Society, 0347(244):1–34, 2007.

[HW69] W. C. Hsiang and C. T. C. Wall.On Homotopy Tori II.Bulletin of the London Mathematical Society, 1(3):341–342, November 1969.

[KS77] Rob C. Kirby and L.C. Siebenmann.Foundational essays on topological manifolds, smoothings, and triangulations.Annals of Mathematics Studies, 88, 1977.

[Pil88] Anand Pillay.On groups and fields definable in o-minimal structures.Journal of Pure and Applied Algebra, 53(3):239–255, 1988.

[Pil04] Anand Pillay.Type-definability, compact lie groups, and o-Minimality.Journal of Mathematical Logic, 4(2):147–162, 2004.

[PS99] Ya’acov Peterzil and Charles Steinhorn.Definable Compactness and Definable Subgroups of o-Minimal Groups.J. London Math. Soc., 59:769–786, 1999.

() May 4, 2011 31 / 32

References IV

[Shi10] Masahiro Shiota.PL and differential topology in o-minimal structure.arXiv:1002.1508, pages 1–48, 2010.

[Str94a] A. Strzebonski.Euler characteristic in semialgebraic and other O-minimal structures.J. Pure and Applied Algebra,, 86:173–201, 1994.

[Str94b] A. Strzebonski.One-dimensional groups in o-minimal structures.J. Pure and Applied Algebra, 96:203–214, 1994.

[vdD98] Lou van den Dries.Tame topology and o-minimal structures, volume 248.LMS Lecture Notes Series, Cambridge Univ. Press, 1998.

() May 4, 2011 32 / 32

topology of de nable groups in tame structuresberardu/art/paris2011/paris.pdf · topology of de...

Documents